Pythagorean Triples and Primes

[Enola Straight]

Pythagorean Triples
http://en.wikipedia.org/wiki/Pythagorean_triples
Scatter plot of the triples
http://en.wikipedia.org/wiki/Image:Pyth ... erplot.jpg
A variation of the Ulam Spiral (scroll down to figure 5)
http://www.numberspiral.com/index.html

I notice a visual similarity between some of the parabolic curves of the Pythagorean triple scatter plot and the spiral curves of the Ulam Spiral.

The obvious curve in fig. 5 represents the Euler polynomial x^2+x+41 prime generator, described on page7 fig.1 as the P+41 curve.

Are the curves in the triple scatter plot and the Ulam Spiral plot mathematically or geometrically related?

[Enola Straight]

Another scatter plot of the triples:
http://www.eddaardvark.co.uk/python_pat ... p6400i.gif

[Enola Straight]

and another one:
http://www.eddaardvark.co.uk/python_pat ... /p6400.gif

[Kuroneko]

I'm not aware of any deep connection between them; although it is possible there is one, I believe it is unlikely because the pattern you speak of is found straightforwardly from considering the nature of Pythagorean triples without any properties of primes. For a Pythagorean triple (x,y,z), x²+y²=z², without loss of generality we may assume that gcd(x,y,z) = 1, for otherwise we can simply factor out the common divisor. Among these primitive Pythagorean triples, it is straightforward to show that x≠y (mod 2), and that all of them with 2|y are generated by
(1) x = u²-v², y = 2uv, z = u²+v², gcd(u,v) = 1, u≠v (mod 2).
I can provide the details of proving this if you're curious, but it's actually more than we need; verifying that the above form generates Pythagorean triples is a matter of basic algebra:
(2) z² = (u²+v²)² = u^4 - 2u²v² + v^4 + 4u²v² = (u²-v²)² + (2uv)² = x² + y².
The fact that all Pythagorean triples are related to a primitive in that form, up to order, is simply an unnecessary bonus that we won't use here.

Now, think of the correspond transformation for (u,v)-plane to the (x,y)-plane, with the coordinates allowed to be real. In other words, let w = u+iv and z = x+iy, so that the Pythagorean transformation (1) is simply z = w². This is a conformal transformation that maps lines to parabolas, except those that pass through the origin. Thus, what the parabolic patterns that you're seeing are generated by straight lines in the (u,v)-plane, with coordinates once again restricted to integers.

---

However, just playing around with Ulam spiral can reveal some curious patterns. On a rectilinear graph, the Ulam spiral has a diagonal of even squares and an opposing diagonal of odd squares (rather than all the squares forming a line as in your link), while the primes tend to form diagonals. So for the diagonals we can refer to an even corner offset of 4n² and some linear term, in general giving a quadratic with leading coefficient 4. Now, by separating even and off cases of x, the Euler polynomial f(x) = x²+x+41 can be seen as a splicing of 4n²-2n+41 and 4n²+2n+41 (let x = 2n, 2n-1). Both of these actually form diagonal patterns, but they aren't too obvious on a standard Ulam spiral, tending to jump between diagonals. However, if we make an Ulam spiral starting with 41 instead of 1, we have something much more visually impressive:

Code: Select all

77 76 75 74 73 72 71 97   Upper-Right: 4n²-2n+41
78 57 56 55 54 53 70 96
79 58 45 44 43 52 69 95
80 59 46 41 42 51 68 94
81 60 47 48 49 50 67 93
82 61 62 63 64 65 66 92
83 84 85 86 87 88 90 91   Lower-Left: 4n²+2n+41

Starting from 41, the upper-right diagonal is desribed by 4n²-2n+41; it runs through the primes
41,43,53,71,97,131,173,223,281,347,421,503,593,691,797,911,1033,1163,1301,1447,1601;
it then sputters on 1763 and 2491, but excepting those the next few terms are also prime:
1763,1933,2111,2297,2491,2693,2903,3121,3347,3581,3823,4073.
The lower-left diagonal, 4n²+2n+41, runs through the primes
41,47,61,83,113,151,197,251,313,383,461,547,641,743,853,971,1097,1231,1373,1523;
going through the non-primes 1681,2021,3233, the rest of the ones listed being primes:
1681,1847,2021,2203,2393,2591,2797,3011,3233,3463,3701,3947,4201,4463,4733,5011,5297,5591.

[Enola Straight]

AH....a reply.

Here is an earlier thread from the Straight Dope:
http://boards.straightdope.com/sdmb/sho ... p?t=432112

Some points raised there could be elaborated upon?



On the scatter plot
http://en.wikipedia.org/wiki/Image:Pyth ... erplot.jpg

We can see one obvious parabolic curve between 1500 on one axis and 3000 on another axis.

Does this curve ( or other parabolic curves) contain points corresponding to the Euler polynomial or other prime generators?

[Kuroneko]

Here is an earlier thread from the Straight Dope:
http://boards.straightdope.com/sdmb/sho ... p?t=432112
Some points raised there could be elaborated upon?

Well, Omphaloskeptic's reply is most informative; translating his point, his (m,n) is my (u,v) and his (a,b) is my (x,y), so he shows that lines of constant m are mapped to parabolas in the (a,b) plane. However, this is actually true in general for all lines in the (m,n) plane, excepting those that pass through the origin, as shown in my previous post.

Does this curve ( or other parabolic curves) contain points corresponding to the Euler polynomial or other prime generators?

There's are plenty of similarities, since both figures are so full of quadratic curves. The point is that the connection is not at all deep, or at least there is no reason to expect it to be deep--any commonality between them is most likely a coincidence.